Rotation Matrices

This page is partly a test of using WordPress LaTeX rendering to generate some math equations, in this case rotation matrices. For now, that’s all it is…

2D

Start with a 2D identity matrix:

$I^2=\begin{bmatrix}1&0\\0&1\end{bmatrix}$

The trick is look at the matrix as two vertical (column) vectors, called i-hat and j-hat. These start off as unit vectors along the x-axis and y-axis, respectively, and the matrix says where they end up.

A matrix describes a transformation of (in this case) the 2D plane. Such a transformation might be a scaling, a rotation, a shear, or some combination. In all cases, the origin stays the same.

In the case of the identity matrix, i-hat ends up at (1,0) and j-hat at (0,1), which is exactly where they started. That’s what makes the identity matrix the identity matrix. Nothing changes!

A 2D rotation matrix looks like this:

$R_z^2(\theta)=\begin{bmatrix}\cos \theta&-\sin \theta\\\sin \theta&\cos \theta\end{bmatrix}$

The name, $R_z^2(\theta)$ identifies it as a Rotation matrix. The superscript indicates that it’s a rotation in two dimensions, and the subscript that the rotation is about the z-axis. Also, R is a function that takes an angle, θ (theta).

The result is a matrix with four values based on the sine and cosine of θ. Given the following table, it’s easy to plug in values for four cases of angle θ:

	0°	90°	180°	270°
sin θ	0	+1	0	-1
cos θ	+1	0	-1	0

So the first rotation matrix, a rotation of zero degrees, looks like this:

$R_z^2(0)=\begin{bmatrix}1&0\\0&1\end{bmatrix}$

Which is the identity matrix. A rotation of zero degrees does nothing.

The other three matrices look like this:

$R_z^2(90)= \begin{bmatrix}0&-1\\+1&0\end{bmatrix}$

$R_z^2(180)= \begin{bmatrix}-1&0\\0&-1\end{bmatrix}$

$R_z^2(270)= \begin{bmatrix}0&+1\\-1&0\end{bmatrix}$

Of course, other angles result in real values.

For instance, a 10-degree rotation matrix looks like this:

$R_z^2(10)=\begin{bmatrix}+0.9848&-0.1736\\+0.1736&+0.9848\end{bmatrix}$

The values are just cos(10) and sin(10) plugged into their appropriate spots.

3D

The 3D identity matrix:

$I^3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$

That’s also the matrix for a rotation of zero about any axis.

The matrix for 3D rotation around the x-axis:

$R_x^3(\theta)=\begin{bmatrix}1&0&0\\0&\cos \theta&-\sin \theta\\0&\sin \theta&\cos \theta\end{bmatrix}$

The three matrices for x-axis rotations of 90°, 180°, and 270°, look like:

$R_x^3(90)=\begin{bmatrix}+1&0&0\\0&0&-1\\0&+1&0\end{bmatrix}$

$R_x^3(180)=\begin{bmatrix}+1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}$

$R_x^3(270)=\begin{bmatrix}+1&0&0\\0&0&+1\\0&-1&0\end{bmatrix}$

The matrix for 3D rotation around the y-axis:

$R_y^3(\theta) = \begin{bmatrix}\cos\theta&0&-\sin\theta\\0&1&0\\\sin\theta&0&\cos\theta\end{bmatrix}$

The matrix for 3D rotation around the z-axis:

$R_z^3(\theta) = \begin{bmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{bmatrix}$

The instances for specific rotations look much like the ones shown above.

4D

The 4D identity matrix:

$I^4=\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$

The matrix for 4D rotation around the Y and Z axes:

$R_{yz}^4(\theta) = \begin{bmatrix}\cos\theta&0&0&-\sin\theta\\0&1&0&0\\0&0&1&0\\\sin\theta&0&0&\cos\theta\end{bmatrix}$

This is the tesseract rotation that seems to move the cubes along the X-axis (the first mode of rotation seen in the video).

The dual-axis rotation allows rotation of all orientations of cubes in 4D space.

In a tesseract, the four cubes with extent in X or W shift between extending in those axes, while the other four cubes (who have both X and W extent), rotate around either their Y or Z axis (depending on which they have).

In a (3D) cube, 4D rotation rotates the X extent into the W dimension and back (reversed), which allows “flipping” a cube. Consider the above matrix in 3D (no W axis):

$R_{x}^3(\theta) = \begin{bmatrix}\cos\theta&0&0\\0&1&0\\0&0&1\end{bmatrix}$

This truncated version shifts the X extent into the W dimension, but that dimension is cut off and not shown. The effect is that the X dimension reduces to zero (at 90°), restores in the negative (reversed) direction (at 180°), reduces to zero (at 270°), and finally restores positively (at 360°).

The matrix for 4D rotation around the X and Z axes:

$R_{xz}^4(\theta) = \begin{bmatrix}1&0&0&0\\0&\cos \theta&0&-\sin \theta\\0&0&1&0\\0&\sin \theta&0&\cos \theta\end{bmatrix}$